Remark 1 Note that, the Hawkes process \(M\) 
itself does not have the Markov property. However, assuming that the
excitation function \(\varphi (\cdot )\)  has an exponential decay kernel, it is possible to prove that the bi-dimensional process \((\lambda , M)\)  is a Markov process (Errais et al. [1]}).
Furthermore, the explosion is avoided by ensuring the condition given by \(\mathbb {E}[B_t] < r\) , (Daw and Pender [2]}).

